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I've seen various bijections between the rationals and the naturals (the first being Cantor's Pairing function (the snake looking one) and another being the Calkin Wilf Tree) many times; but I can always remember being slightly disturbed by the lack of 'naturality' in the bijection (before I even knew formally how to express 'naturality'); in some sense the bijections always seemed a bit arbitrary and not inherently part of the structure of the rationals/the naturals.

Recently I've been exposed to some notion of 'naturality' in Category Theory, and loosely I have come to understand a 'natural' bijection as one that in some sense requires no choice of any kind, and arises inherently from the structures involved. Is there a way of talking about the 'naturality' of the bijections between the rationals and the naturals? I'm not even sure which Categories are at play in this scenario, or even if this is a well defined question or not.

Björn Friedrich
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QCD_IS_GOOD
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    Taking a walk and had some sort of vague idea; so the rationals come about by localizing (that could be the wrong word) the naturals, i.e. introducing inverses for all elements, this sort of process should be a functor on the category of rings, call it L. We also have a forgetful functor to the category of sets, call it F. So, we are saying something about... actually I'm not sure whether this is the right train of thought – QCD_IS_GOOD May 25 '17 at 11:16
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    Not an answer, but like you till now I considered the tree you talk about "weird" to say the least, but the interpretation brought by Wolfgang in his resistors post https://math.stackexchange.com/questions/2160766/how-many-resistors-are-needed/2206744#2206744 was enlighting. In fact each rational appear as a linear chain of serial, parallel resistors of 1 Ohm, this way the Stern-Brocot tree looks way more natural. – zwim May 25 '17 at 11:18
  • A natural bijection is just an isomorphism between Set-valued functors. So here's a natural bijection: consider functors $\mathbb{N},\mathbb{Q} : 1\to\mathbf{Set}$ which pick out the sets of natural and rational numbers respectively. Take any bijection between those sets. That bijection is then (the sole component of) a natural bijection between the functors. Obviously this is not interesting. For it to become interesting, you need to decide what structure you want to preserve e.g. via a Lawvere theory. – Derek Elkins left SE May 25 '17 at 14:09
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    @JoshuaLin: In my reading, naturality is just kind of synonym for functoriality, in that, an $Ob,{\bf A}\to Mor{\bf B}$ assignment becomes a natural transformation iff it extends to a functor ${\bf A}\to {\bf B}^{\to}$ where ${\bf B}^{\to}$ is the arrow category. Note that, your first comment is aiming exactly the functoriality. – Berci May 25 '17 at 22:48
  • An interesting thing is that in the bijections you mentioned there aren't any specific construction of $\mathbb{N} $ or of $\mathbb{Q} $, only their properties as initial objects in some category. So, in some way, there is already some categorical aspect in play. – Grassy LittleRoot May 28 '17 at 13:07

2 Answers2

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It is a common misconception that a natural isomorphism (in the categorical sense) is an isomorphism $A \to B$ between two single objects $A,B$ which is natural in some sense. This is not true. The concept of a natural isomorphism applies to functors between two categories. In other words, a natural isomorphism is really a bunch of isomorphisms $F(x) \to G(x)$, one for each object $x$ of the domain of $F$ and $G$. Of course, any object can be regarded as a constant functor, and then any isomorphism between two objects leads to a natural isomorphism between the associated functors, but this is rather boring and has nothing to do with naturality in the common sense.

Your question is about isomorphisms between $\mathbb{N}$ and $\mathbb{Q}$ in the category of sets. [At this point, let me mention that there is no isomorphism between these objects in more interesting categories such as commutative monoids, ordered sets, topological spaces, etc.] As I've said, it does not make much sense to speak of natural isomorphisms here, because we have no functors given. But it is possible to consider the following two functors:

Consider the category $\mathcal{C}$ of semirings with isomorphisms of semirings, and the category $\mathcal{D}$ of sets and isomorphisms of sets. There is a forgetful functor $F : \mathcal{C} \to \mathcal{D}$, and there is a functor $G : \mathcal{C} \to \mathcal{D}$ which maps a semiring to the underlying set of the total ring of fractions of its Grothendieck ring (given by adjoining additive inverses and then multiplicative inverses by regular elements). Thus, $F(\mathbb{N},+,\cdot)=\mathbb{N}$ and $G(\mathbb{N},+,\cdot)=\mathbb{Q}$. Now one might ask if there is a natural isomorphism $F \to G$. But this is not the case.

HeinrichD
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  • After asking my question I was having the same kind of sentiment; I realized by 'naturality' I needed to talk about it in a 'larger setting' (i.e. the functors you mentioned) similar to how we can say that a vector space is naturally isomorphic to its double dual by considering a natural transformation between the double dual functor and the identity functor – QCD_IS_GOOD May 28 '17 at 10:18
  • But I also had some strange thoughts about "zooming in", I.e. Considering things on a smaller scale; perhaps functors from the category of natural numbers (with morphisms as less than or equal to? Considering the naturals as a poset?) and the category of rationals (as a poset?) Is this a meaningful idea or is it barking up the wrong tree? – QCD_IS_GOOD May 28 '17 at 10:19
  • Waht do you want to prove/disprove, specifically? – HeinrichD May 28 '17 at 10:27
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In an earlier post, I have described a bijection between $\mathbb{N}$ and $\mathbb{Q}$ which I consider somewhat less artificial than e.g. Cantor's pairing function. Perhaps it is something like this you are looking for?

jpvee
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